Classical and Quantum Initial Value Problems for Models of Chronology Violation

TL;DR

This paper investigates classical and quantum initial value problems in models with chronology violation, revealing existence, uniqueness, and nonunitarity issues, and contrasting with path integral approaches.

Contribution

It provides a detailed analysis of classical and quantum solutions in chronology-violating spacetimes, highlighting differences from previous path integral methods and exploring quantum-classical correspondence.

Findings

Classical solutions exist and are unique in weak regimes, but not in strong coupling.

Quantum evolution is unitary for two-point models but nonunitary in general.

Quantum theory can select a classical limit even with multiple classical solutions.

Abstract

We study the classical and quantum theory of a class of nonlinear differential equations on chronology violating spacetime models in which space consists of finitely many discrete points. Classically, in the linear and weakly nonlinear regimes (for generic choices of parameters) we prove existence and uniqueness of solutions corresponding to initial data specified before the dischronal region; however, uniqueness (but not existence) fails in the strongly coupled regime. The evolution preserves the symplectic structure. The quantum theory is approached via the quantum initial value problem (QIVP); that is, by seeking operator-valued solutions to the equation of motion with initial data representing the canonical (anti)commutation relations. Using normal operator ordering, we construct solutions to the QIVP for both Bose and Fermi statistics (again for generic choice of parameters) and prove that these solutions are unique. For models with two spatial points, the resulting evolution is unitary; however, for a more general model the evolution fails to preserve the (anti)commutation relations and is therefore nonunitary. We show that this nonunitary evolution cannot be described using a superscattering operator with the usual properties. We present numerical evidence to show that the bosonic quantum theory can pick out a unique classical limit for certain ranges of the coupling strength, even when there are many classical solutions. We show that the quantum theory depends strongly on the choice of operator ordering. In addition, we show that our results differ from those obtained using the ``self-consistent path integral''. It follows that the path integral evolution does not correspond to a solution of the equation of motion.